# American Institute of Mathematical Sciences

September  2011, 3(3): 313-322. doi: 10.3934/jgm.2011.3.313

## Euler equations on a semi-direct product of the diffeomorphisms group by itself

 1 Institute for Applied Mathematics, University of Hanover, D-30167 Hanover, Germany 2 School of Mathematical Science, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland 3 LATP, CNRS & University of Provence, 39 Rue F. Joliot-Curie, 13453 Marseille Cedex 13

Received  August 2011 Revised  September 2011 Published  November 2011

The geodesic equations of a class of right invariant metrics on the semi-direct product $Diff(\mathbb{S}^1)$Ⓢ$Diff(\mathbb{S}^1)$ are studied. The equations are explicitly described, they have the form of a system of coupled equations of Camassa-Holm type and possess singular (peakon) solutions. Their integrability is further investigated, however no compatible bi-Hamiltonian structures on the corresponding dual Lie algebra $(Vect(\mathbb{S}^1)$Ⓢ$Vect(\mathbb{S}^1))^{*}$ are found.
Citation: Joachim Escher, Rossen Ivanov, Boris Kolev. Euler equations on a semi-direct product of the diffeomorphisms group by itself. Journal of Geometric Mechanics, 2011, 3 (3) : 313-322. doi: 10.3934/jgm.2011.3.313
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